 noticed this problem, OK. Now that... I show you that studying F1 is enough to understand the troubles. And then I quickly mentioned the work of waitingINAUDIBLE. After waiting letting was also Arnold, Arnold produced this work when it was 22, I guess, I guess you are aging... maybe you are older than you. You are still late for... It's a bit depressing, you see, the age of dispersal. When they produce this masterpiece. OK, so now consider the most... climate situation. Let me assume that this stuff... Let me assume that this is... analytic. So admit a... where m is an integer. m is a vector with... m1, m2, mn. And each of these are integers. And so this is Fourier series on the angle. And I assume that this direct system go to zero exponentially fast. So a very, very, very, very, very climate assumption. And then I write the same... I try to look this in the same form. Let me use the same data. And I put this here. And I compute this. I have to compute this quantity. OK. And... OK, this is simple. So I have to... So I have h0, f1. This is the derivative h0 theta n f1 respect to i minus... This is zero. This is zero. This is omega. OK, this is omega. And so I have that... Oh, sorry. Let me put it here in minus. So I have this one. So the minus is this. And so I have some... Omega n i. And here I have this. So I use this formula. So I have some on m. When I perform the derivative, the derivative is only here. The derivative I have i, i, m, n. You see? I, m, n. And then I have f1, m. OK. This is the Poisson bracket. Then I have to repeat the same with the other term. And then compare... So let me... Let me consider the case where f0 is a certain action. So I have the minus, the derivative of f0, minus the sum of n, n, derivative of f0, respect to delta n, derivative of this, respect to i, n, plus the sum of n, f0, respect to i, n. This is zero. This is zero. This term is zero. This is delta. This is delta. This is delta. Because this is delta, j, n. And here... Here I have... Sorry. It's the same computation before. Here I have sum of m. When I derive this, I have h1, m, i, a, i, m, theta. And the derivative... I have an i, m, derivative of i, m, j. So I have to compare this with this. So I have an equation for f1. I hope that everything is correct. So what is... The important thing is this part. This part. So I have to compare this and what I have. So apparently I have the solution because I have f1, m, i, equal to what? Equal h1, m, i, and... We have some decoration. Some decoration is not important. What is important is that... I have to divide by this term. This is the important term. Sum on n omega n, i, m, n. This is the important stuff. And apparently I solved the problem. This is the problem of the denominator. So you know this. You know this. You know this. So apparently you solved the problem. But now you have to be sure that this is possible to do. There are two possibilities. That omega is rational. The frequency are rational among them. Or are irrational. If they are rational, there is a problem. But you can say, this is a rare number. The mathematicians say rational. Who care rational? Rational are small. Are... Rare in the sense of the back measure. So, yeah. The situation, omega... The omega rational. No. But this you can decide that you don't consider the case with the rational omega. So the omega are irrational. So if are irrational, apparently you are happy because this is for sure different from zero. It's true that it's different from zero. But it can be arbitrary small. Because there are... But you have that. This is different from zero. But arbitrary small. So you can have some combination of m that this can be arbitrary small. And this is the end of the story. This is the end of the story. This approach cannot work because you have formally the solution. But the solution is meaningless because this can be arbitrary large. At least for certain m and certain region of the phase space. This is called small denominator problem. Small denominator this denominator which not can be small, but is small. For sure is small. For some value this is small. And this is the end of the story. So apparently this is terrible new. This is terrible new for all the people working in celestial mechanics. But apparently is a wonderful new. For us it means people working in statistical mechanics because in statistical mechanics is a wonderful result there is apparently a very positive consequence in statistical mechanics. Because if you have a conservation law for sure the system is not ergodic. This is clear. Now here you have not a conservation law. The result is the good direction. Maybe you can expect that since you have no conservation law then the system is ergodic. Ergodic is what you can hope. This idea is tempting. Even a genius like Rigo Fermi arrived to this wrong conclusion. Even Rigo Fermi arrived to the wrong conclusion which actually is wrong. And it is wrong for a very subtle reason. And the person who realized so you see that this result for the people working in celestial mechanics is very negative. People working in statistical mechanics apparently is positive. The situation is in between. It is in between the sense that the result is not so negative for the celestial mechanics but it is not so positive for statistical mechanics. So at the end both the community complain. As it is said in Italy mal comune mezzo gaudio. As it is said in English. For the Italian. OK. Now the rest of the story. Kolmogorov. Kolmogorov. This result was in blatant contrast with the fact that the people working in celestial mechanics perform in a naive way the perturbative computation with also rather good result. But apparently this result of one creation this is forbidden. In principle this is forbidden. But using this formal method of the expert of celestial mechanics the result are not so bad. And the reason that is not so bad has been understood by Kolmogorov. The reason is the following that this result this statement is that doesn't exist a conservation law in the whole phase space but it's not clear if it's possible to have a sort of local existence in the phase space. The idea is roughly there is the following. You see that this thing says OK apart in the region where this is more so you say if I remove the region where this is more maybe I can go on. It's not a joke to do this because you have to remove the part and you have to be sure that the part you remove is not the whole space. So this is the idea of Kolmogorov. Kolmogorov was able to prove the following. Kolmogorov and then the part on the motor. The following result. So you have H0I plus epsilon A1 H is one. OK. There is some hypothesis for example you have the hypothesis you have the omega the omega which are given by this stuff and you have the matrix determinant of this A must be different from zero and then you have some some regularity some regularity on this. OK. Some regularity on this. You have that. You have the following result. That. Four. Small. OK. So if epsilon equals zero you have the the tori. So the tori means I n equal constant and theta nt equal zero plus omega nt. OK. So you have the the motion involved on this tori like in donuts. Then you can then when epsilon is smaller when epsilon is smaller according to in buvankere this tori are destroyed. Right. But are destroyed generically but the fact that in a region let me call I don't know omega omega epsilon the region of omega epsilon the I the one is in order epsilon the formation of the tori for epsilon equals zero. OK. So it means that you imagine that you have this tori. These were epsilon equals zero. Then when you introduce epsilon instead of this you have something like that. But this not everywhere in a certain region. OK. Then the integral part of the work is that the measure of this region go to one where epsilon go to zero. This is an integral part. This is an integral part. We sense that OK. And this is so this means that there are always some situation where this is not true but there are a lot part of the phase where this is true. So this is this is the anonite OK. You have this OK. You have this OK. You have this this amytonian system with amytonian system you have that no no OK. You have this amytonian system. You see this is the integral part you are just two oscillator and then you have anonite integral part some cubic stuff and so according to then and then you you study this with the Poincaré map I guess Fabio introduced with the Poincaré map and so with the Poincaré map you understand if the system is regular or not OK. So for example this is here this depends on the energy. OK. If the energy is small then you have that the perturbation is small so you have that in the grid part of the phase space you have regular behavior this curve you see these are the the intersection of the so you have this sort of tube then you cut and you serve this OK. This means that you are looking these are the survived torai but there are some regions where this is not true you have this this is a unique trajectory this is chaotic region so you have that 99% of the region is regular but there are 1, 2% which is chaotic OK. Then of course this depends on the epsilon so in fact if you increase epsilon if you increase epsilon so you have that still remains some region but now are not so so big so if you increase again if you increase energy you have that so almost the full space is chaotic but remain still so you see that from the statistical statistical mechanics point of view these are disturbing these are disturbing in the area you don't want this OK. Why for people working in other field don't want this OK. This is the story. So you see that at the variance with the dissipative system in the dissipative system you have a sort of a transition chaos not in the Lorentz system so up to a certain control parameter value the system is somehow trivial and then after this critical value you have chaos in Hamiltonian system is different in Hamiltonian system you have no critical value of epsilon this is very this means that what happened increasing immediately the system is not is not integrable immediately and if small if epsilon is small you have that only in small region you see the trouble this is the difference the fact that in this case you have known attractor so you have known attractor this means that your result depends on the point where you start this is not true in dissipative system in dissipative system you start from any point after a certain transient you approach to an attractor here is different so this is a big difference between dissipative and Hamiltonian system infact the two communities are more or less split so the people who are experts in Hamiltonian system people who are experts in dissipative system ok now but let me so now let me just explain you that the Hamiltonian system usually the Hamiltonian system the first thing you have a system with bolts with springs these things this is true but there are very nice application of Hamiltonian system in a different contest for example in fluid dynamics let me explain why so imagine that you want to follow particle in a fluid so you have a fluid you have a fluid with a certain a certain velocity field ok let me consider the case in 2D in 2D ok and in compressive fluid since it is compressive this means that you have you have two components you have two components but you have a constraint this constraint so you understand that in some sense the two components cannot be independent ok and so one possibility is just to use one of the components but for tradition the people introduce what is called the string function such that u1 is equal u2 is equal to minus so you see that this is automatically this is true so this is the way to remove one of the two components instead of to decide to use u1 or 22 you use another quantity string function ok this is the elementary market now consider the following problem consider the problem that you put the dust in your in your field and you want to follow this dust ok this dust must be small enough in such a way you don't feel it ok but not too small otherwise you feel the brave emotion ok and this is actually what what happened in the experiment in Jofici the people put the balloons in the atmosphere what is this consider the boy the boy consider the boy the boys the boys in the in the ocean and so they follow so you study you study this equation ok this is the the question of the dust a small dust in the fluid follow are affected by the fluid ok and because of this so this means that we are x x equal x dot equal u1 y dot equal u2 but this is the derivative of this this is minus the derivative of this so you remember that if you have q dot equal derivative of h respect to p p dot equal minus derivative respect to q you have the same structure so you have the same structure instead instead we have q p h you have x y c ok so you have usually we consider the case where h is autonomous but now we allow h to be time dependent ok is not forbidden and why because it is the unique interest situation because in the case c is is autonomous the system is integrable the system integrable because p is conserved ok so now the you have that the do you want to study this problem with c equal a part which is stationary and a part which is time dependent particular periodic in time this for example when you imagine you want to study the dust how evolve so for example contaminant you want to be contaminant you have the velocity you have velocity field which change in time and you want to understand how this contaminant ok this problem this this problem is called a system with one degree of freedom and alpha why because one but then alpha because there is also the time and this is this is the first non-trivial problem you can try to understand in a Victorian system why and because in order to understand where this is the the problem you can reformulate the problem in term of simplicity maps and so for example you can instead to look at x continuous time if this is periodic period let me call t I can define let me find x, n and y, n like this x at time n, t and of course of course I can I can write these two these two term as a function let me call this gamma gamma n so I have that gamma n plus 1 must be a certain function of gamma n of course because the system is deterministic and I look at any period so this is since the starting problem is a Newtonian this problem must be simplistic so you have a conservation of of the area so the fact that you have is let me put it here also epsilon ok when epsilon equals 0 where epsilon equals 0 you have a conservative system sorry you have an integrable system then where epsilon is different from 0 you have a in an integrable system you want to understand how this behaves ok this is at theoretical level is the simplest nontrivial question you can you can wonder ok even these have been studied by Poincari always Poincari ok and it introduces the idea of which is extremely important and powerful the idea of homoclinic intersection I don't want to discuss in the tables I have no time but just to give you what it means so imagine but it is necessary to pass technically to pass for the imagine that you have pendulum just pendulum you have pendulum you have p squared over 2 minus cosine of p ok you have something like that and so forget the constant so you have something like that so this is the potential ok this is the potential this is the potential so you see that if you are at this energy you have a periodic motion so you have relations cosine of q q q sorry and this is the potential and and so if you look at the qp so here you have something like that ok and if you are here you have if you are here you have something like that you have open line you have the this is the oscillation and vibration ok if you are exactly at this energy if you are exactly at this energy you have if you are exactly at this energy you have something like that you have something like that these two points coincide of course you have exactly like that so tzv.��� hill wall Dynamic highlight.ion v predovitec, to je je sought-up. Pa da ga se jažem, ja stabal,FF oj. Vskega, srečna, Produkti no naprej, 4-e Mojaff italija, Co zelo? Osoba je...... you see you have this particular trajectory which is the which separates the... rotation from... libration from oscillation, if you are here, that doesn't... you don't see nothing particular. If you are here you don't see anything particular, In, kako jaz sem tukaj, tukaj je tukaj kandidej, kaj je kao. In, kako jaz sem tukaj, kaj jaz sem tukaj, kaj jaz sem tukaj, kaj jaz sem tukaj. Tukaj, ni tako vzadno v izgledu. To je vzadno. To je tukaj, kaj se kaj so pri aimsku, pa ta job delna vzadno אני. To je vzadno, ki je tako vzadno, ki je vzadno, ki je vzadno. Ki jaz sem dolgova, ki jaz sem dolgova, ki jaz sem dolgova, ki jaz sem dolgova, ki jaz sem dolgova. Zato se počim vidimo, ki jaz sem dolgova ozvr triječe. In vidiš, da vse predstavljeni, da vse predstavljeni, vse predstavljeni kaos. To je poslednje, da poživajte. To je nekaj izgled, nekaj poživajte. In kako se zelo, nekaj nekaj vse predstavljeni, nekaj, ta kombinacija kajos in regulari. Zame, kako imam tukaj pravda? Zato je inštačnja aplikacija v geofiziku. Zato je inštačnja aplikacija v geofiziku. Zato v geofiziku je... v geofiziku je nekaj tukaj. In tukaj je tukaj večo mekanizma, kaj je tukaj opočen. Če, čas čas smo se zazaj, čas si posledajte, je pri vse z pošličenje v vso vseh stacijnega vseh vseh stacija, čas je, čas je, čas je, čas je vseh stacija, čas je vseh stacija. V zelo v sežite, v naturalnje stvari se počkaj vseči se vseči začne. A zato počkaj se vseči izvah vseči na odličnji vseči. Prezat, zato vseči je zelo potrožen. Zato je bilo vseči vseči odličnji vseči začnji. Zato se počkaj se izvah vseči. In, izgledaj, to je zdravitko, bo to je laminarstv. To je laminarstv. Laminarstv je očen, da se vse zelo v del, ali, da se vse zelo v del, se zelo v kaos. To je neko nekaj idejo, ki je Arno, neko neko Arno, in ki je vse zelo vse zelo vse zelo, ali, nekaj, zelo, zelo, zelo, zato sem zelo vzelo vse zelo, x dot equal u x t. Zato vedimo, da, even if this vector field is smooth, simple, apparently simple, you can have chaos, why not? No, but if you translate this in terms of fluid dynamics, apparently it is a paradox, because you have that field, when you look at the field, it is very smooth, very regular, then when you put dust, dust performs strange behavior. Which apparently is a paradoxical situation, but if you look from the point of view of dynamical system, it's not paradoxical at all. This remark is due to Arnold many, many years ago, and actually you can have that even very, very simple velocity field, just with sine, cosine, and so on, can produce astonishingly complicated Lagrangian transport. This, for example, the people working in fluid dynamics, working in geophysics, they, the idea of Arnold was in 66, something like that. And geophysics upset this idea at least 20 years later, maybe even 30. The sum, in Italy, very few geophysicists accept this idea, because they refuse to enter in two mathematical tools. OK, I guess I can stop here. Tomorrow we will have this written exam. Of course it will be not on this part. So my suggestion, look at the maps, look at Peron Frobenius, look at the Godissi, the exam will not be too serious. OK, thanks. Do you have questions? So this last part, I understand, is a bit too dense, but just to give you an idea that dynamical system is not just a game of simple stuff, no, we can enter in real physics. OK, this is just to mention that there are real applications, for example, in geophysics. OK, so thank you very much, Angelo.